Select the correct answer.

Jane is saving her money in order to purchase a new racing bike. She initially saves $3 and plans to double the amount she saves each month. The bike Jane wants is $1,536 at the local bike shop.

Which equation represents this situation, and after how many months, $t$, will Jane have enough money to purchase the bike?
A. $3(2)^t=1,536 ; t=11$
B. $3(1.2)^t=1,536: t=35$
C. $(3
cdot 2)^t=1,536 ; t=9$
D. $3(2)^t=1,536 ; t=9

Question

Select the correct answer.

Jane is saving her money in order to purchase a new racing bike. She initially saves $3 and plans to double the amount she saves each month. The bike Jane wants is $1,536 at the local bike shop.

Which equation represents this situation, and after how many months, $t$, will Jane have enough money to purchase the bike?
A. $3(2)^t=1,536 ; t=11$
B. $3(1.2)^t=1,536: t=35$
C. $(3 
cdot 2)^t=1,536 ; t=9$
D. $3(2)^t=1,536 ; t=9

Anonymous · Answer

The equation representing Jane's savings is 3 ( 2 ) t = 1 , 536 . She will have enough money to purchase the bike after t = 9 months. Therefore, the correct answer is option D. 
 ;

GinnyAnswer · Answer

Set up the equation representing Jane's savings: 3 × 2 t = 1536 . 
 Divide both sides by 3: 2 t = 512 . 
 Express 512 as a power of 2: 512 = 2 9 . 
 Solve for t : t = 9 . The correct answer is D. 3 ( 2 ) t = 1 , 536 ; t = 9 . 
 
 Explanation 
 
 Problem Analysis Jane starts with $3 and doubles her savings each month. We need to find the equation that models this situation and determine how many months it will take for her to save $1 , 536 . 
 
 Setting up the Equation The equation representing Jane's savings is given by: 3 × 2 t = 1536 where t is the number of months. 
 
 Isolating the Exponential Term To solve for t , we first divide both sides of the equation by 3: 3 3 × 2 t ​ = 3 1536 ​ 2 t = 512 
 
 Expressing 512 as a Power of 2 Now, we need to express 512 as a power of 2. We know that: 512 = 2 9 
 
 Solving for t So, our equation becomes: 2 t = 2 9 Since the bases are equal, the exponents must be equal: t = 9 
 
 Final Answer Therefore, it will take Jane 9 months to save enough money to purchase the bike. The correct equation is 3 ( 2 ) t = 1 , 536 , and t = 9 .

Examples 
 Exponential growth, like Jane's savings, is a common phenomenon. Imagine a population of bacteria that doubles every hour. If you start with 3 bacteria, the equation 3 ( 2 ) t models the population after t hours. This type of calculation is crucial in fields like biology, finance (compound interest), and even computer science (algorithm complexity). Understanding exponential growth helps in predicting outcomes and making informed decisions.

Answer (2)

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